Question

Consider a laundry business – called Care Cleaners – that uses strong chemicals in its dry cleaning process. Some of the excess chemicals used by Care Cleaners find their way into a local water source and impose costs on a nearby brewery – called Brewsters – that uses the water in its brewing process. Both businesses make decisions to maximize their profits.

The price of a unit of laundry is $20 and the price of a unit of beer is $16. The quantity of laundry is denoted as L and the quantity of beer is denoted as B. The total cost of producing laundry is L². The total cost of producing beer is LB + B².

[4] What is the efficient quantity of Laundry and Beer? Hint: efficiency requires that joint profits are maximized.

[5] Now, suppose that the Laundry business has the right to keep polluting. According to the Coase Theorem, describe a bargain between both businesses that could lead to the efficient solution from [4]. In particular, describe which firm makes the payment and how much they offer.  

Answer 1

The price of a unit of laundry is $20 and the price of a unit of beer is $16. The quantity of laundry is denoted as L and the quantity of beer is denoted as B. The total cost of producing laundry is L². The total cost of producing beer is LB + B² [GIVEN]

(a) For efficiency, we need to maximize the total profit of the two firms when they are operating together , ie., we maximise joint profits ( Total revenue - Total Cost)

max _(L,B) 20L + 16B - (L²) - (LB + B²)

differentiating with respect to L first, we get    20 - 2L - B = 0 -- (1)

now, differentiating with respect to B, we get 16 - L - 2B = 0 -- (2)

Solving -- (1) and (2) , we get L = 8 and B = 4

Therefore, the efficient quantity of Laundry is 8 units and that of Beer is 4 units.

Suppose they don't operate together (This will help in in part b)

If only the Laundry operates (zero abatement of pollution), max_L 20L - L² . differentiation wrt L , we get 20 - 2L = 0 => L =10    -- (3)

If only the Brewery operates, max_B 16B - (B² ) . differentiation wrt B , we get 16 - 2B = 0 => B = 8   -- (4)

(b) Now, Laundry has the right to pollute , let's consider the profit of the Brewery which is the victim of the pollution.

if both firms operate, the brewery will have to pay to the Laundry firm to reduce pollution. When both operate, abatement of pollution will be at level of quantity found in part (a) ie at L = 8 and B = 4. The brewery will have to pay for the abatement, which is the reduction in profits of the Laundry firm in moving to this abatement level from abatement level of zero. The payment made to the Laundry firm will be equal to : profit of Laundry at 10 units of Laundry - profit of Laundry at 8 units of Laundry]

= [20(10) - (10)²]- [20(8) - (8)²] = 100 - 96 = 4

Therefore, the Brewery pays $4 to the Laundry firm